In an interview with Charlie Rose, Warren Buffett discussed the mortgage crisis and said “beware of geeks bearing formulas.” It turns out that you don’t need to know any math to understand where these formulas go wrong.

There are a whole host of Greek letters that represent investing formulas that few people understand. We have alpha, beta, sigma, and five other letters that are associated with stock options. Even people who rely on these formulas often don’t know much about them. A trader may tell you that he buys stocks when one or more Greek letters have certain values, but this is different from understanding what the numbers mean.

One thing the formulas have in common is that they are usually calculated based on recent history of equity prices. So, any investing decision based on these formulas is necessarily a reaction to recent events. Often this results in traders gambling that the near future to be similar to the recent past.

This sort of behaviour is similar to the gambler who bets on the outcome of a coin toss. A string of heads makes some gamblers bet on heads because they think the coin is “hot,” while other gamblers think that tails is “due.” Of course, they’re both wrong. A fair coin has a 50/50 chance of coming up heads or tails regardless of what has happened on recent tosses.

An Example

To illustrate what can go wrong with formulas, let’s look at an example. Suppose an investment called a “Dollar Bond” pays a dollar every trading day. However, each day there is also a 1 in 50 chance that it will be a bad day where 5% of the remaining bonds disappear. So, if you own 100 of these bonds, you’ll get $100 each day. When a bad day happens, you’ll suddenly only have only 95 Dollar Bonds and will get $95 per day until the next bad day.

To estimate a fair price for Dollar Bonds, let’s assume that the risk-free interest rate is 4% per year. To give an expected return of 4% (assuming 250 trading days per year), the price of Dollar Bonds works out to $864 each. Of course, Dollar Bonds have volatility, and investors would actually want to pay less than $864 depending on their level of risk aversion.

In the real world, we don’t actually know the exact probabilities of bad outcomes. If we were to use the same principal as the Greek letter formulas, we would estimate the chances of a bad day by examining the number of bad days in the past.

This leads us to my own Greek letter formula that I’ll call iota (because you shouldn’t give it an iota of respect as we’ll see). To calculate iota, we take the number of bad days over the past year and use that to come up with an estimate of the value of a Dollar bond that gives an expected 4% return.

I simulated a 20-year history for Dollar Bonds giving the following results:

For something that is supposed to have a steady value of $864, the iota measure sure jumps around a lot. In fact, iota jumped to over $6000 briefly around year 7. This is because there happened to be a period of just over one year with no bad days. So, for a short time, iota thought that bad days never happen. Pity the poor investor who bought Dollar Bonds then and promptly lost most of his money.

Of course, any math guy worth his salt would look at this chart and change the way iota is calculated to make its output smoother. But that doesn’t mean that the new measure would necessarily be any better. It would just seem better.

Formulas named after Greek letters tend to seem impressive, and we give them more respect than they deserve. It’s dangerous to use any formula if you don’t understand how it works and what its limitations are. I use mathematical analysis every chance I get, but not as a replacement for thinking.

To quote John Maynard Keynes, “it is better to be roughly right than precisely wrong.”

Excellent example. One always has to ask, "what are the assumptions?". Usually the model is pretty reasonable but getting the assumptions that conform to the real world is hard, if not impossible. Currently, I find myself struggling with what assumptions to make about stock market returns. Do we use the last 50, 90, 200 or 1000 years as the input data? Do we include only the best - the USA - and exclude the rest of the world?

ReplyDeleteWow. Awesome example. It should be in textbooks.

ReplyDeleteCanadian Investor and Patrick:

ReplyDeleteThanks for the kind words. I had fun creating this example. I agree that it is hard to decide what assumptions to make about future returns. I tend to believe that in the long term, owners will beat loaners (stocks will beat bonds), but I don't know what to assume about the gap.